Gradient flows of the entropy for jump processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 920-945.

On considère une nouvelle distance entre les mesures de probabilité sur n . Elle est construite à partir d’un processus de saut par une variante non-locale de la formule de Benamou-Brenier. Pour les processus de Lévy on démontre que le semigroupe engendré par l’opérateur non-local associé est le flot de gradient de l’entropie par rapport à la nouvelle distance. On démontre aussi que l’entropie est convexe le long des géodésiques dans ce cas.

We introduce a new transport distance between probability measures on d that is built from a Lévy jump kernel. It is defined via a non-local variant of the Benamou-Brenier formula. We study geometric and topological properties of this distance, in particular we prove existence of geodesics. For translation invariant jump kernels we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance and show that the entropy is convex along geodesics.

DOI : 10.1214/12-AIHP537
Classification : 60J75, 35S10, 45K05, 49J45, 60G51
Mots clés : jump process, Lévy process, gradient flow, entropy, optimal transport
@article{AIHPB_2014__50_3_920_0,
     author = {Erbar, Matthias},
     title = {Gradient flows of the entropy for jump processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {920--945},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {3},
     year = {2014},
     doi = {10.1214/12-AIHP537},
     mrnumber = {3224294},
     zbl = {1311.60091},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/12-AIHP537/}
}
TY  - JOUR
AU  - Erbar, Matthias
TI  - Gradient flows of the entropy for jump processes
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2014
SP  - 920
EP  - 945
VL  - 50
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/12-AIHP537/
DO  - 10.1214/12-AIHP537
LA  - en
ID  - AIHPB_2014__50_3_920_0
ER  - 
%0 Journal Article
%A Erbar, Matthias
%T Gradient flows of the entropy for jump processes
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2014
%P 920-945
%V 50
%N 3
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/12-AIHP537/
%R 10.1214/12-AIHP537
%G en
%F AIHPB_2014__50_3_920_0
Erbar, Matthias. Gradient flows of the entropy for jump processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 920-945. doi : 10.1214/12-AIHP537. http://www.numdam.org/articles/10.1214/12-AIHP537/

[1] L. Ambrosio, N. Gigli and G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition. Lectures in Mathematics. Birkhäuser, Basel, 2008. | MR | Zbl

[2] L. Ambrosio, N. Gigli and G. Savaré. Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Preprint, 2011. Available at arXiv:1106.2090. | MR

[3] L. Ambrosio, G. Savaré and L. Zambotti. Existence and stability for Fokker-Planck equations with log-concave reference measure. Probab. Theory Related Fields 145 (2009) 517-564. | MR | Zbl

[4] D. Applebaum. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 93. Cambridge Univ. Press, Cambridge, 2004. | MR | Zbl

[5] D. Bakry and M. Émery. Diffusions hypercontractives. In Séminaire de Probabilités XIX 177-206. Lecture Notes in Math. 1123. Springer, Berlin, 1985. | Numdam | MR | Zbl

[6] M. Barlow, R. Bass, Z.-G. Chen and M. Kassmann. Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc. 361 (2009) 1963-1999. | MR | Zbl

[7] J.-D. Benamou and Y. Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375-393. | MR | Zbl

[8] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl

[9] G. Buttazzo. Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Research Notes in Mathematics Series. Longman Scientific and Technical, Harlow, 1989. | MR | Zbl

[10] L. Caffarelli and L. Silvestre. The Evans-Krylov theorem for nonlocal fully nonlinear equations. Ann. of Math. (2) 174 (2011) 1163-1187. | MR | Zbl

[11] Z.-Q. Chen and T. Kumagai. Heat kernel estimates for stable-like processes on d-sets. Stochastic Process. Appl. 108 (2003) 27-62. | MR | Zbl

[12] S.-N. Chow, W. Huang, Y. Li and H. Zhou. Fokker-Planck equations for a free energy functional or Markov process on a graph. Arch. Ration. Mech. Anal. 203 (2012) 969-1008. | MR | Zbl

[13] S. Daneri and G. Savaré. Eulerian calculus for the displacement convexity in the Wasserstein distance. SIAM J. Math. Anal. 40 (2008) 1104-1122. | MR | Zbl

[14] J. Dolbeault, B. Nazaret and G. Savaré. A new class of transport distances between measures. Calc. Var. Partial Differential Equations 34 (2009) 193-231. | MR | Zbl

[15] M. Erbar. The heat equation on manifolds as a gradient flow in the Wasserstein space. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 1-23. | Numdam | MR | Zbl

[16] M. Erbar and J. Maas. Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal. 206 (2012) 997-1038. | MR | Zbl

[17] S. Fang, J. Shao and K.-Th. Sturm. Wasserstein space over the Wiener space. Probab. Theory Related Fields 146 (2010) 535-565. | MR | Zbl

[18] N. Gigli. On the heat flow on metric measure spaces: Existence, uniqueness and stability. Calc. Var. Partial Differential Equations 39 (2010) 101-120. | MR | Zbl

[19] N. Gigli, K. Kuwada and S.-I. Ohta. Heat flow on Alexandrov spaces. Comm. Pure Appl. Math. 66 (2013) 307-331. | MR | Zbl

[20] R. Jordan, D. Kinderlehrer and F. Otto. The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1-17. | MR | Zbl

[21] J. Lott and C. Villani. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169 (2009) 903-991. | MR | Zbl

[22] J. Maas. Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261 (2011) 2250-2292. | MR | Zbl

[23] R. Mccann. A convexity principle for interacting gases. Adv. Math. 128 (1997) 153-179. | MR | Zbl

[24] A. Mielke. Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Partial Differential Equations 48 (2013) 1-31. | MR | Zbl

[25] S.-I. Ohta and K.-Th. Sturm. Heat flow on Finsler manifolds. Comm. Pure Appl. Math. 62 (2009) 1386-1433. | MR | Zbl

[26] F. Otto. The geometry of dissipative evolution equations: The porous medium equation. Comm. Partial Differential Equations 26 (2001) 101-174. | MR | Zbl

[27] F. Otto and C. Villani. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000) 361-400. | MR | Zbl

[28] K.-Th. Sturm. On the geometry of metric measure spaces. I. Acta Math. 196 (2006) 65-131. | MR | Zbl

[29] K.-Th. Sturm. On the geometry of metric measure spaces. II. Acta Math. 196 (2006) 133-177. | MR | Zbl

[30] C. Villani. Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften 338. Springer, Berlin, 2009. | MR | Zbl

Cité par Sources :